The chronological age used in demography describes the linear evolution of the life of a living being. The chronological age cannot give precise information about the exact developmental stage or aging processes an organism has reached. On the contrary, the biological age (or epigenetic age) represents the true evolution of the tissues and organs of the living being. Biological age is not always linear and sometimes proceeds by discontinuous jumps. These jumps can be negative (we then speak of rejuvenation) or positive (in the event of premature aging), and they can be dependent on endogenous events such as pregnancy (negative jump) or stroke (positive jump) or exogenous ones such as surgical treatment (negative jump) or infectious disease (positive jump). The article proposes a mathematical model of the biological age by defining a valid model for the two types of jumps (positive and negative). The existence and uniqueness of the solution are solved, and its temporal dynamic is analyzed using a moments equation. We also provide some individual-based stochastic simulations.
Citation: Jacques Demongeot, Pierre Magal. Population dynamics model for aging[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 19636-19660. doi: 10.3934/mbe.2023870
[1] | Minlong Lin, Ke Tang . Selective further learning of hybrid ensemble for class imbalanced increment learning. Big Data and Information Analytics, 2017, 2(1): 1-21. doi: 10.3934/bdia.2017005 |
[2] | Subrata Dasgupta . Disentangling data, information and knowledge. Big Data and Information Analytics, 2016, 1(4): 377-390. doi: 10.3934/bdia.2016016 |
[3] | Qinglei Zhang, Wenying Feng . Detecting Coalition Attacks in Online Advertising: A hybrid data mining approach. Big Data and Information Analytics, 2016, 1(2): 227-245. doi: 10.3934/bdia.2016006 |
[4] | Tieliang Gong, Qian Zhao, Deyu Meng, Zongben Xu . Why Curriculum Learning & Self-paced Learning Work in Big/Noisy Data: A Theoretical Perspective. Big Data and Information Analytics, 2016, 1(1): 111-127. doi: 10.3934/bdia.2016.1.111 |
[5] | Xin Yun, Myung Hwan Chun . The impact of personalized recommendation on purchase intention under the background of big data. Big Data and Information Analytics, 2024, 8(0): 80-108. doi: 10.3934/bdia.2024005 |
[6] | Pankaj Sharma, David Baglee, Jaime Campos, Erkki Jantunen . Big data collection and analysis for manufacturing organisations. Big Data and Information Analytics, 2017, 2(2): 127-139. doi: 10.3934/bdia.2017002 |
[7] | Zhen Mei . Manifold Data Mining Helps Businesses Grow More Effectively. Big Data and Information Analytics, 2016, 1(2): 275-276. doi: 10.3934/bdia.2016009 |
[8] | Ricky Fok, Agnieszka Lasek, Jiye Li, Aijun An . Modeling daily guest count prediction. Big Data and Information Analytics, 2016, 1(4): 299-308. doi: 10.3934/bdia.2016012 |
[9] | M Supriya, AJ Deepa . Machine learning approach on healthcare big data: a review. Big Data and Information Analytics, 2020, 5(1): 58-75. doi: 10.3934/bdia.2020005 |
[10] | Sunmoo Yoon, Maria Patrao, Debbie Schauer, Jose Gutierrez . Prediction Models for Burden of Caregivers Applying Data Mining Techniques. Big Data and Information Analytics, 2017, 2(3): 209-217. doi: 10.3934/bdia.2017014 |
The chronological age used in demography describes the linear evolution of the life of a living being. The chronological age cannot give precise information about the exact developmental stage or aging processes an organism has reached. On the contrary, the biological age (or epigenetic age) represents the true evolution of the tissues and organs of the living being. Biological age is not always linear and sometimes proceeds by discontinuous jumps. These jumps can be negative (we then speak of rejuvenation) or positive (in the event of premature aging), and they can be dependent on endogenous events such as pregnancy (negative jump) or stroke (positive jump) or exogenous ones such as surgical treatment (negative jump) or infectious disease (positive jump). The article proposes a mathematical model of the biological age by defining a valid model for the two types of jumps (positive and negative). The existence and uniqueness of the solution are solved, and its temporal dynamic is analyzed using a moments equation. We also provide some individual-based stochastic simulations.
For a continuous risk outcome
Given fixed effects
In this paper, we assume that the risk outcome
y=Φ(a0+a1x1+⋯+akxk+bs), | (1.1) |
where
Given random effect model (1.1), the expected value
We introduce a family of interval distributions based on variable transformations. Probability densities for these distributions are provided (Proposition 2.1). Parameters of model (1.1) can then be estimated by maximum likelihood approaches assuming an interval distribution. In some cases, these parameters get an analytical solution without the needs for a model fitting (Proposition 4.1). We call a model with a random effect, where parameters are estimated by maximum likelihood assuming an interval distribution, an interval distribution model.
In its simplest form, the interval distribution model
The paper is organized as follows: in section 2, we introduce a family of interval distributions. A measure for tail fatness is defined. In section 3, we show examples of interval distributions and investigate their tail behaviours. We propose in section 4 an algorithm for estimating the parameters in model (1.1).
Interval distributions introduced in this section are defined for a risk outcome over a finite open interval
Let
Let
Φ:D→(c0,c1) | (2.1) |
be a transformation with continuous and positive derivatives
Given a continuous random variable
y=Φ(a+bs), | (2.2) |
where we assume that the range of variable
Proposition 2.1. Given
g(y,a,b)=U1/(bU2) | (2.3) |
G(y,a,b)=F[Φ−1(y)−ab]. | (2.4) |
where
U1=f{[Φ−1(y)−a]/b},U2=ϕ[Φ−1(y)] | (2.5) |
Proof. A proof for the case when
G(y,a,b)=P[Φ(a+bs)≤y] |
=P{s≤[Φ−1(y)−a]/b} |
=F{[Φ−1(y)−a]/b}. |
By chain rule and the relationship
∂Φ−1(y)∂y=1ϕ[Φ−1(y)]. | (2.6) |
Taking the derivative of
∂G(y,a,b)∂y=f{[Φ−1(y)−a]/b}bϕ[Φ−1(y)]=U1bU2. |
One can explore into these interval distributions for their shapes, including skewness and modality. For stress testing purposes, we are more interested in tail risk behaviours for these distributions.
Recall that, for a variable X over (−
For a risk outcome over a finite interval
We say that an interval distribution has a fat right tail if the limit
Given
Recall that, for a Beta distribution with parameters
Next, because the derivative of
z=Φ−1(y) | (2.7) |
Then
Lemma 2.2. Given
(ⅰ)
(ⅱ) If
(ⅲ) If
Proof. The first statement follows from the relationship
[g(y,a,b)(y1−y)β]−1/β=[g(y,a,b)]−1/βy1−y=[g(Φ(z),a,b)]−1/βy1−Φ(z). | (2.8) |
By L’Hospital’s rule and taking the derivatives of the numerator and the denominator of (2.8) with respect to
For tail convexity, we say that the right tail of an interval distribution is convex if
Again, write
h\left(z, a, b\right) = \mathrm{log}\left[g\left(\mathrm{\Phi }\left(z\right), a, b\right)\right], | (2.9) |
where
g\left(y, a, b\right) = \mathrm{exp}\left[h\left(z, a, b\right)\right]. | (2.10) |
By (2.9), (2.10), using (2.6) and the relationship
{g}_{y}^{'} = {[h}_{z}^{'}\left(z\right)/{\rm{ \mathsf{ ϕ} }}\left(\mathrm{z}\right)]\mathrm{e}\mathrm{x}\mathrm{p}[h({\mathrm{\Phi }}^{-1}\left(y\right), a, b)], \\ {g}_{yy}^{''} = \left[\frac{{h}_{zz}^{''}\left(z\right)}{{{\rm{ \mathsf{ ϕ} }}}^{2}\left(\mathrm{z}\right)}-\frac{{h}_{z}^{'}\left(z\right){{\rm{ \mathsf{ ϕ} }}}_{\mathrm{z}}^{'}\left(z\right)}{{{\rm{ \mathsf{ ϕ} }}}^{3}\left(\mathrm{z}\right)}+\frac{{h}_{\mathrm{z}}^{\mathrm{'}}\left(\mathrm{z}\right){h}_{\mathrm{z}}^{\mathrm{'}}\left(\mathrm{z}\right)}{{{\rm{ \mathsf{ ϕ} }}}^{2}\left(\mathrm{z}\right)}\right]\mathrm{e}\mathrm{x}\mathrm{p}\left[h\right({\mathrm{\Phi }}^{-1}\left(y\right), a, b) ]. | (2.11) |
The following lemma is useful for checking tail convexity, it follows from (2.11).
Lemma 2.3. Suppose
In this section, we focus on the case where
One can explore into a wide list of densities with different choices for
A.
B.
C.
D.D.
Densities for cases A, B, C, and D are given respectively in (3.3) (section 3.1), (A.1), (A.3), and (A5) (Appendix A). Tail behaviour study is summarized in Propositions 3.3, 3.5, and Remark 3.6. Sketches of density plots are provided in Appendix B for distributions A, B, and C.
Using the notations of section 2, we have
By (2.5), we have
\mathrm{log}\left(\frac{{U}_{1}}{{U}_{2}}\right) = \frac{{-z}^{2}+2az-{a}^{2}+{b}^{2}{z}^{2}}{2{b}^{2}} | (3.1) |
= \frac{{-\left(1-{b}^{2}\right)\left(z-\frac{a}{1-{b}^{2}}\right)}^{2}+\frac{{b}^{2}}{1-{b}^{2}}{a}^{2}}{2{b}^{2}}\text{.} | (3.2) |
Therefore, we have
g\left(\mathrm{y}, a, b\right) = \frac{1}{b}\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{{-\left(1-{b}^{2}\right)\left(z-\frac{a}{1-{b}^{2}}\right)}^{2}+\frac{{b}^{2}}{1-{b}^{2}}{a}^{2}}{2{b}^{2}}\right\}\text{.} | (3.3) |
Again, using the notations of section 2, we have
g\left(y, p, \rho \right) = \sqrt{\frac{1-\rho }{\rho }}\mathrm{e}\mathrm{x}\mathrm{p}\{-\frac{1}{2\rho }{\left[{\sqrt{1-\rho }{\mathrm{\Phi }}^{-1}\left(y\right)-\mathrm{\Phi }}^{-1}\left(p\right)\right]}^{2}+\frac{1}{2}{\left[{\mathrm{\Phi }}^{-1}\left(y\right)\right]}^{2}\}\text{, } | (3.4) |
where
Proposition 3.1. Density (3.3) is equivalent to (3.4) under the relationships:
a = \frac{{\Phi }^{-1}\left(p\right)}{\sqrt{1-\rho }} \ \ \text{and}\ \ b = \sqrt{\frac{\rho }{1-\rho }}. | (3.5) |
Proof. A similar proof can be found in [19]. By (3.4), we have
g\left(y, p, \rho \right) = \sqrt{\frac{1-\rho }{\rho }}\mathrm{e}\mathrm{x}\mathrm{p}\{-\frac{1-\rho }{2\rho }{\left[{{\mathrm{\Phi }}^{-1}\left(y\right)-\mathrm{\Phi }}^{-1}\left(p\right)/\sqrt{1-\rho }\right]}^{2}+\frac{1}{2}{\left[{\mathrm{\Phi }}^{-1}\left(y\right)\right]}^{2}\} |
= \frac{1}{b}\mathrm{exp}\left\{-\frac{1}{2}{\left[\frac{{\Phi }^{-1}\left(y\right)-a}{b}\right]}^{2}\right\}\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{1}{2}{\left[{\mathrm{\Phi }}^{-1}\left(y\right)\right]}^{2}\right\} |
= {U}_{1}/{(bU}_{2}) = g(y, a, b)\text{.} |
The following relationships are implied by (3.5):
\rho = \frac{{b}^{2}}{1{+b}^{2}}, | (3.6) |
a = {\Phi }^{-1}\left(p\right)\sqrt{1+{b}^{2}}\text{.} | (3.7) |
Remark 3.2. The mode of
\frac{\sqrt{1-\rho }}{1-2\rho }{\mathrm{\Phi }}^{-1}\left(p\right) = \frac{\sqrt{1+{b}^{2}}}{1-{b}^{2}}{\mathrm{\Phi }}^{-1}\left(p\right) = \frac{a}{1-{b}^{2}}. |
This means
Proposition 3.3. The following statements hold for
(ⅰ)
(ⅱ)
(ⅲ) If
Proof. For statement (ⅰ), we have
Consider statement (ⅱ). First by (3.3), if
{\left[g\left(\mathrm{\Phi }\left(\mathrm{z}\right), a, b\right)\right]}^{-1/\beta } = {b}^{1/\beta }\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{{\left({b}^{2}-1\right)z}^{2}+2az-{a}^{2}}{2\beta {b}^{2}}) | (3.8) |
By taking the derivative of (3.8) with respect to
-\left\{\partial {\left[g\left(\mathrm{\Phi }\left(\mathrm{z}\right), a, b\right)\right]}^{-\frac{1}{\beta }}/\partial z\right\}/{\rm{ \mathsf{ ϕ} }}\left(\mathrm{z}\right) = \sqrt{2\pi }{b}^{\frac{1}{\beta }}\frac{\left({b}^{2}-1\right)z+a}{\beta {b}^{2}}\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{{\left({b}^{2}-1\right)z}^{2}+2az-{a}^{2}}{2\beta {b}^{2}}+\frac{{z}^{2}}{2})\text{.} | (3.9) |
Thus
\left\{\partial {\left[g\left(\mathrm{\Phi }\left(\mathrm{z}\right), a, b\right)\right]}^{-\frac{1}{\beta }}/\partial z\right\}/{\rm{ \mathsf{ ϕ} }}\left(\mathrm{z}\right) = -\sqrt{2\pi }{b}^{\frac{1}{\beta }}\frac{\left({b}^{2}-1\right)z+a}{\beta {b}^{2}}\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{{\left({b}^{2}-1\right)z}^{2}+2az-{a}^{2}}{2\beta {b}^{2}}+\frac{{z}^{2}}{2})\text{.} | (3.10) |
Thus
For statement (ⅲ), we use Lemma 2.3. By (2.9) and using (3.2), we have
h\left(z, a, b\right) = \mathrm{log}\left(\frac{{U}_{1}}{{bU}_{2}}\right) = \frac{{-\left(1-{b}^{2}\right)\left(z-\frac{a}{1-{b}^{2}}\right)}^{2}+\frac{{b}^{2}}{1-{b}^{2}}{a}^{2}}{2{b}^{2}}-\mathrm{l}\mathrm{o}\mathrm{g}\left(b\right)\text{.} |
When
Remark 3.4. Assume
li{m}_{z⤍+\infty }-\left\{{\partial \left[g\left(\mathrm{\Phi }\left(\mathrm{z}\right), a, b\right)\right]}^{-\frac{1}{\beta }}/\partial z\right\}/{\rm{ \mathsf{ ϕ} }}\left(\mathrm{z}\right) |
is
For these distributions, we again focus on their tail behaviours. A proof for the next proposition can be found in Appendix A.
Proposition 3.5. The following statements hold:
(a) Density
(b) The tailed index of
Remark 3.6. Among distributions A, B, C, and Beta distribution, distribution B gets the highest tailed index of 1, independent of the choices of
In this section, we assume that
First, we consider a simple case, where risk outcome
y = \mathrm{\Phi }\left(v+bs\right), | (4.1) |
where
Given a sample
LL = \sum _{i = 1}^{n}\left\{\mathrm{log}f\left(\frac{{z}_{i}-{v}_{i}}{b}\right)-\mathrm{l}\mathrm{o}\mathrm{g}{\rm{ \mathsf{ ϕ} }}\left({z}_{i}\right)-logb\right\}\text{, } | (4.2) |
where
Recall that the least squares estimators of
SS = \sum _{i = 1}^{n}{({z}_{i}-{v}_{i})}^{2} | (4.3) |
has a closed form solution given by the transpose of
{\rm{X}} = \left\lceil {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {1\;\;{x_{11}} \ldots {x_{k1}}}\\ {1\;\;{x_{12}} \ldots {x_{k2}}} \end{array}}\\ \ldots \\ {1\;\;{x_{1n}} \ldots {x_{kn}}} \end{array}} \right\rceil , {\rm{Z}} = \left\lceil {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{z_1}}\\ {{z_2}} \end{array}}\\ \ldots \\ {{z_n}} \end{array}} \right\rceil . |
The next proposition shows there exists an analytical solution for the parameters of model (4.1).
Proposition 4.1. Given a sample
Proof. Dropping off the constant term from (4.2) and noting
LL = -\frac{1}{2{b}^{2}}\sum _{i = 1}^{n}{({z}_{i}-{v}_{i})}^{2}-nlogb, | (4.4) |
Hence the maximum likelihood estimates
Next, we consider the general case of model (1.1), where the risk outcome
y = \mathrm{\Phi }[v+ws], | (4.5) |
where parameter
(a)
(b)
Given a sample
LL = \sum _{i = 1}^{n}-{\frac{1}{2}[\left({z}_{i}-{v}_{i}\right)}^{2}/{w}_{i}^{2}-{u}_{i}], | (4.6) |
LL = \sum _{i = 1}^{n}\{-\left({z}_{i}-{v}_{i}\right)/{w}_{\mathrm{i}}-2\mathrm{log}[1+\mathrm{e}\mathrm{x}\mathrm{p}[-({z}_{i}-{v}_{i})/{w}_{i}]-{u}_{i}\}, | (4.7) |
Recall that a function is log-concave if its logarithm is concave. If a function is concave, a local maximum is a global maximum, and the function is unimodal. This property is useful for searching maximum likelihood estimates.
Proposition 4.2. The functions (4.6) and (4.7) are concave as a function of
Proof. It is well-known that, if
For (4.7), the linear part
In general, parameters
Algorithm 4.3. Follow the steps below to estimate parameters of model (4.5):
(a) Given
(b) Given
(c) Iterate (a) and (b) until a convergence is reached.
With the interval distributions introduced in this paper, models with a random effect can be fitted for a continuous risk outcome by maximum likelihood approaches assuming an interval distribution. These models provide an alternative regression tool to the Beta regression model and fraction response model, and a tool for tail risk assessment as well.
Authors are very grateful to the third reviewer for many constructive comments. The first author is grateful to Biao Wu for many valuable conversations. Thanks also go to Clovis Sukam for his critical reading for the manuscript.
We would like to thank you for following the instructions above very closely in advance. It will definitely save us lot of time and expedite the process of your paper's publication.
The views expressed in this article are not necessarily those of Royal Bank of Canada and Scotiabank or any of their affiliates. Please direct any comments to Bill Huajian Yang at h_y02@yahoo.ca.
[1] |
E. Bernabeu, D. L. McCartney, D. A. Gadd, R. F. Hillary, A. T. Lu, L. Murphy, et al., Refining epigenetic prediction of chronological and biological age, Genome Med., 5 (2023), 1–15. https://doi.org/10.1186/s13073-023-01161-y doi: 10.1186/s13073-023-01161-y
![]() |
[2] |
P. Jain, A. M Binder, B. Chen, H. Parada, L. C. Gallo, J. Alcaraz, et al., Analysis of epigenetic age acceleration and healthy longevity among older US women, JAMA Network Open, 5 (2022), e2223285–e2223285. https://doi.org/10.1001/jamanetworkopen.2022.23285 doi: 10.1001/jamanetworkopen.2022.23285
![]() |
[3] |
J. R. Poganik, B. Zhang, G. S. Baht, A. Tyshkovskiy, A. Deik, C. Kerepesi, et al., Biological age is increased by stress and restored upon recovery, Cell Metabol., 35 (2023), 807–820. https://doi.org/10.1016/j.cmet.2023.03.015 doi: 10.1016/j.cmet.2023.03.015
![]() |
[4] |
J. Demongeot, Biological boundaries and biological age, Acta Biotheoret., 27 (2009), 397–418. https://doi.org/10.1007/s10441-009-9087-8 doi: 10.1007/s10441-009-9087-8
![]() |
[5] | D. Applebaum, Lévy processes and stochastic calculus, Cambridge university press,, 2009. https://doi.org/10.1017/CBO9780511809781 |
[6] | B. Ycart, Modéles et algorithmes markoviens, volume 39. Springer Science & Business Media, Berlin, Heidelberg, 2002. |
[7] |
H. X. Huang, M. A. Milevsky, T. S. Salisbury, Retirement spending and biological age, J. Econom. Dynam. Control, 84 (2017), 58–76. https://doi.org/10.1016/j.jedc.2017.09.003 doi: 10.1016/j.jedc.2017.09.003
![]() |
[8] |
R. Siddiqui, S. Maciver, A. Elmoselhi, N. C. Soares, N. A. Khan, Longevity, cellular senescence and the gut microbiome: Lessons to be learned from crocodiles, Heliyon, 7 (2021). https://doi.org/10.1016/j.heliyon.2021.e08594 doi: 10.1016/j.heliyon.2021.e08594
![]() |
[9] |
P. Talukder, A. Saha, S. Roy, G. Ghosh, D. Dutta Roy, S. Barua, Progeria—a rare genetic condition with accelerated ageing process, Appl. Biochem. Biotechnol., 195 (2023), 2587–2596. https://doi.org/10.1007/s12010-021-03514-y doi: 10.1007/s12010-021-03514-y
![]() |
[10] |
L. N. Nguyen, T. Kanneganti, PANoptosis in viral infection: The missing puzzle piece in the cell death field, J. Molecular Biol., 434 (2022), 167249. https://doi.org/10.1016/j.jmb.2021.167249 doi: 10.1016/j.jmb.2021.167249
![]() |
[11] |
D. S. Knopman, S. D. Edland, R. H. Cha, R. C. Petersen, W. A. Rocca, Incident dementia in women is preceded by weight loss by at least a decade, Neurology, 69 (2007), 739–746. https://doi.org/10.1212/01.wnl.0000267661.65586.33 doi: 10.1212/01.wnl.0000267661.65586.33
![]() |
[12] | L. Hayflick, The serial cultivation of human diploid cell strains, Nephrol. Dialys. Transplant., 11 (1996), 1822–1824. |
[13] |
Y. J. Kim, H. S. Kim, Y. R. Seo, Genomic approach to understand the association of dna repair with longevity and healthy aging using genomic databases of oldest-old population, Oxidat. Med. Cellular Longev., 2018. https://doi.org/10.1155/2018/2984730 doi: 10.1155/2018/2984730
![]() |
[14] |
E. R. Stead, I. Bjedov, Balancing dna repair to prevent ageing and cancer, Exper. Cell Res., 405 (2021), 112679. https://doi.org/10.1016/j.yexcr.2021.112679 doi: 10.1016/j.yexcr.2021.112679
![]() |
[15] |
E. Kashdan, S. Bunimovich-Mendrazitsky, Hybrid discrete-continuous model of invasive bladder cancer, Math. Biosci. Eng., 10 (2013), 729–742. https://doi.org/10.3934/mbe.2013.10.729 doi: 10.3934/mbe.2013.10.729
![]() |
[16] |
J. A Sherratt, J. D. Murray, Models of epidermal wound healing, Proceedings of the Royal Society of London, Series B: Biological Sciences, 241 (1990), 29–36. https://doi.org/10.1098/rspb.1990.0061 doi: 10.1098/rspb.1990.0061
![]() |
[17] |
P. Gerdhem, K. A. M. Ringsberg, K. Åkesson, K. J. Obrant, Clinical history and biologic age predicted falls better than objective functional tests, J. Clin. Epidemiol., 58 (2005), 226–232. https://doi.org/10.1016/j.jclinepi.2004.06.013 doi: 10.1016/j.jclinepi.2004.06.013
![]() |
[18] |
J. W. Shay, W. E. Wright, Hayflick, his limit, and cellular ageing, Nat. Rev. Molecular Cell Biol., 1 (2000), 72–76. https://doi.org/10.1038/35036093 doi: 10.1038/35036093
![]() |
[19] |
J. Jylhävä, N. L. Pedersen, S. Hägg, Biological age predictors, EBioMedicine, 521 (2017), 29–36. https://doi.org/10.1016/j.ebiom.2017.03.046 doi: 10.1016/j.ebiom.2017.03.046
![]() |
[20] | D. A Sinclair, M. D. LaPlante, Lifespan: Why We Age—and Why We Don't Have To, Atria books,, 2019. |
[21] | G. F. Webb, Theory of nonlinear age-dependent population dynamics, CRC Press,, 1985. |
[22] |
H. R. Thieme, Semiflows generated by lipschitz perturbations of non-densely defined operators, Differ. Integral Equat., 3 (1990), 1035–1066. https://doi.org/10.57262/die/1379101977 doi: 10.57262/die/1379101977
![]() |
[23] | T. Cazenave, A. Haraux, An introduction to semilinear evolution equations, volume 13. Oxford Lecture Mathematics and,, 1998. |
[24] | K.-J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, volume 194. Springer, New York, 2000. |
[25] | P. Magal, S. G. Ruan, Theory and applications of abstract semilinear Cauchy problems, volume 201. Springer, New York, 2018. https://doi.org/10.1007/978-3-030-01506-0 |
[26] | M. Z. Darabseh, T. M. Maden-Wilkinson, G. Welbourne, R. C. I. Wüst, N. Ahmed, H. Aushah, et al., Fourteen days of smoking cessation improves muscle fatigue resistance and reverses markers of systemic inflammation, Sci. Rep., 11 (2021), 12286. |
[27] |
K. Hijazi, B. Malyszko, K. Steiling, X. H. Xiao, G. Liu, Y. O. Alekseyev, et al., Tobacco-related alterations in airway gene expression are rapidly reversed within weeks following smoking-cessation, Sci. Rep., 9 (2019), 6978. https://doi.org/10.1038/s41598-019-43295-3 doi: 10.1038/s41598-019-43295-3
![]() |
[28] |
T. R. Schlam, M. E. Piper, J. W. Cook, M. C. Fiore, T. B. Baker, Life 1 year after a quit attempt: Real-time reports of quitters and continuing smokers, Ann. Behav. Med., 44 (2012), 309–319. https://doi.org/10.1007/s12160-012-9399-9 doi: 10.1007/s12160-012-9399-9
![]() |
[29] |
B. W. Heckman, J. Dahne, L. J. Germeroth, A. R. Mathew, E. J. Santa Ana, M. E. Saladin, M. J. Carpenter, Does cessation fatigue predict smoking-cessation milestones? a longitudinal study of current and former smokers, J. Consult. Clin. Psychol., 865 (2018), 903. https://doi.org/10.1037/ccp0000338 doi: 10.1037/ccp0000338
![]() |
[30] | E. Mortaz, M. R. Masjedi, I. Rahman, Outcome of smoking cessation on airway remodeling and pulmonary inflammation in copd patients, Tanaffos, 10 (2011), 7. |
[31] | United States Public Health Service Office of the Surgeon General; National Center for Chronic Disease Prevention and Health Promotion (US) Office on Smoking and Health. Smoking Cessation: A Report of the Surgeon General [Internet]. Washington (DC): US Department of Health and Human Services; 2020.. https://www.ncbi.nlm.nih.gov/books/NBK555591/. |
[32] |
M. Houston, Stopping smoking could speed recovery after operations, BMJ, 327 (2003), 360. https://doi.org/10.1136/bmj.327.7411.360-d doi: 10.1136/bmj.327.7411.360-d
![]() |
[33] |
T. Napso, H. E. J. Yong, J. Lopez-Tello, A. N. Sferruzzi-Perri, The role of placental hormones in mediating maternal adaptations to support pregnancy and lactation, Front. Physiol., 9 (2018), 1091. https://doi.org/10.3389/fphys.2018.01091 doi: 10.3389/fphys.2018.01091
![]() |
[34] |
I. M. Conboy, M. J. Conboy, A. J. Wagers, E. R. Girma, I. L. Weissman, T. A Rando, Rejuvenation of aged progenitor cells by exposure to a young systemic environment, Nature, 433 (2005), 760–764. https://doi.org/10.1038/nature03260 doi: 10.1038/nature03260
![]() |
[35] |
T. F. Michaeli, N. Laufer, J. Y. Sagiv, A. Dreazen, Z. Granot, E. Pikarsky, et al., The rejuvenating effect of pregnancy on muscle regeneration, Aging Cell, 14 (2015), 698–700. https://doi.org/10.1111/acel.12286 doi: 10.1111/acel.12286
![]() |
[36] |
A. H. Shadyab, M. L. S. Gass, M. L. Stefanick, M. E. Waring, C. A Macera, L. C. Gallo, et al., Maternal age at childbirth and parity as predictors of longevity among women in the united states: The women's health initiative, Am. J. Public Health, 107 (2017), 113–119. https://doi.org/10.2105/AJPH.2016.303503 doi: 10.2105/AJPH.2016.303503
![]() |
[37] |
C. P. Ryan, M. G. Hayes, N. R. Lee, T. W. McDade, M. J. Jones, M. S. Kobor, et al., Reproduction predicts shorter telomeres and epigenetic age acceleration among young adult women, Sci. Rep., 8 (2018), 11100. https://doi.org/10.1038/s41598-018-29486-4 doi: 10.1038/s41598-018-29486-4
![]() |
[38] |
J. Lima-Júnior, L. C. M. Arruda, M. de Oliveira, K. C. R. Malmegrim, Thymus rejuvenation after autologous hematopoietic stem cell transplantation in patients with autoimmune diseases, Thymus Transcript.Cell Biol., (2019), 295–309. https://doi.org/10.1007/978-3-030-12040-5_14 doi: 10.1007/978-3-030-12040-5_14
![]() |
[39] |
F. Nakagawa, R. K. Lodwick, C. J. Smith, R. Smith, V. Cambiano, J. D. Lundgren, et al., Projected life expectancy of people with HIV according to timing of diagnosis, Aids, 26 (2012), 335–343. https://doi.org/10.1097/QAD.0b013e32834dcec9 doi: 10.1097/QAD.0b013e32834dcec9
![]() |